Kneser ranks of random graphs and minimum difference representations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10367120" target="_blank" >RIV/00216208:11320/17:10367120 - isvavai.cz</a>

Výsledek na webu

<a href="http://www.sciencedirect.com/science/article/pii/S1571065317301646?via%3Dihub" target="_blank" >http://www.sciencedirect.com/science/article/pii/S1571065317301646?via%3Dihub</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.endm.2017.06.079" target="_blank" >10.1016/j.endm.2017.06.079</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Kneser ranks of random graphs and minimum difference representations

Popis výsledku v původním jazyce

Every graph $G=(V,E)$ is an induced subgraph of some Kneser graph of rank $k$, i.e., there is an assignment of (distinct) $k$-sets $v mapsto A_v$ to the vertices $vin V$ such that $A_u$ and $A_v$ are disjoint if and only if $uvin E$. The smallest such $k$ is called the {em Kneser rank} of $G$ and denoted by $fKn(G)$. As an application of a result of Frieze and Reed concerning the clique cover number of random graphs we show that for constant $0&lt; p&lt; 1$ there exist constants $c_i=c_i(p)&gt;0$, $i=1,2$ such that with high probability [ c_1 n/(log n)&lt; fKn(G) &lt; c_2 n/(log n). ] We apply this to other graph representations defined by Boros, Gurvich and Meshulam. A {em $k$-min-difference representation} of a graph $G$ is an assignment of a set $A_i$ to each vertex $iin V(G)$ such that $ ijin E(G) ,, Leftrightarrow , , min {|A_isetminus A_j|,|A_jsetminus A_i| }geq k. $ The smallest $k$ such that there exists a $k$-min-difference representation of $G$ is denoted by $f_{min}(G)$. Balogh and Prince proved in 2009 that for every $k$ there is a graph $G$ with $f_{min}(G)geq k$. We prove that there are constants $c&apos;&apos;_1, c&apos;&apos;_2&gt;0$ such that $c&apos;&apos;_1 n/(log n)&lt; f_{min}(G) &lt; c&apos;&apos;_2n/(log n)$ holds for almost all bipartite graphs $G$ on $n+n$ vertices.

Název v anglickém jazyce

Kneser ranks of random graphs and minimum difference representations

Popis výsledku anglicky

Every graph $G=(V,E)$ is an induced subgraph of some Kneser graph of rank $k$, i.e., there is an assignment of (distinct) $k$-sets $v mapsto A_v$ to the vertices $vin V$ such that $A_u$ and $A_v$ are disjoint if and only if $uvin E$. The smallest such $k$ is called the {em Kneser rank} of $G$ and denoted by $fKn(G)$. As an application of a result of Frieze and Reed concerning the clique cover number of random graphs we show that for constant $0&lt; p&lt; 1$ there exist constants $c_i=c_i(p)&gt;0$, $i=1,2$ such that with high probability [ c_1 n/(log n)&lt; fKn(G) &lt; c_2 n/(log n). ] We apply this to other graph representations defined by Boros, Gurvich and Meshulam. A {em $k$-min-difference representation} of a graph $G$ is an assignment of a set $A_i$ to each vertex $iin V(G)$ such that $ ijin E(G) ,, Leftrightarrow , , min {|A_isetminus A_j|,|A_jsetminus A_i| }geq k. $ The smallest $k$ such that there exists a $k$-min-difference representation of $G$ is denoted by $f_{min}(G)$. Balogh and Prince proved in 2009 that for every $k$ there is a graph $G$ with $f_{min}(G)geq k$. We prove that there are constants $c&apos;&apos;_1, c&apos;&apos;_2&gt;0$ such that $c&apos;&apos;_1 n/(log n)&lt; f_{min}(G) &lt; c&apos;&apos;_2n/(log n)$ holds for almost all bipartite graphs $G$ on $n+n$ vertices.

Druh

J_SC - Článek v periodiku v databázi SCOPUS

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GJ16-01602Y" target="_blank" >GJ16-01602Y: Topologické a geometrické přístupy k permutačním třídám a grafovým vlastnostem</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Electronic Notes in Discrete Mathematics

ISSN

1571-0653

e-ISSN

—

Svazek periodika

61

Číslo periodika v rámci svazku

August

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

5

Strana od-do

499-503

Kód UT WoS článku

—

EID výsledku v databázi Scopus

2-s2.0-85026787337